How polygamy is proven depends on the laws of the state regarding bigamy and polygamy. But the government does have a way of tracking and stopping multiple marriages. When you apply for a marriage license the state will do a background search to see if you have ever applied for a marriage license before and if previous marriages have been legally annulled or not. If you are still legally married to someone, a marriage license will not be issued to you. This doesn’t stop people from living polygamy however, they simply don’t get legally married to more than one person. Instead they will have a marriage ceremony that is not legally recognized, such as a religious or spiritual ceremony.
In addition, you can only have one “surviving” spouse in your Social Security records. (Although ex-spouses can apply under your benefits if they meet certain requirements.)